Cantor Confusion
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From: mueckenh on 27 Nov 2006 09:11 Virgil schrieb: > In article <1164447079.521499.79580(a)l39g2000cwd.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > Virgil schrieb: > > > > > > > > > > For finite indexes it is correct. > > > > > > > > > > Infinite things are not constrained to behave in all respects like > > > > > finite ones. That is because "infinite" means "not finite". > > > > > > > > But *the lines are all finite*. > > > > > > The *set* of lines in WM's so-called EIT is not finite. > > > > The number of elements of every line is finite. And only that is > > relevant. > > WM is wrong again. In considering anything related to ALL lines, the > finiteness or infiniteness of the set of all lines is critical. > Any consideration of the "diagonal", which is built from the ends of ALL > lines, So the diagonal is not outside of any line? So there is a finite line as long as the diagonal? > > > > > > > That's just why I chose the EIT as > > > > example. > > > > > > > So your magic belief is easily > > > > destroyed. > > > > > > Not by those who cannot keep their quantifiers straight. > > > > In dealing with linearly ordered finite sets there is no quantifier > > magic. > > There is considerable quantifier logic involved, No, it is not logic but a mental disease. Regards, WM
From: mueckenh on 27 Nov 2006 09:30 Dik T. Winter schrieb: > In article <1164445719.457858.15120(a)l39g2000cwd.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > > > In article <1164281187.964067.115190(a)m7g2000cwm.googlegroups.com> muecken= > > h(a)rz.fh-augsburg.de writes: > > > ... > > > > Cantor wrote: "It is remarkable that removing a countable set leaves > > > > the plaine *being connected*." But it is not remarkable that removing > > > > an uncountable set leaves the plaine being connected? What a foolish > > > > assertion. > > > > > > But removing an uncountable set can leave the plane either connected or > > > disconnected. What is remarkable about that? > > > > Nothing. In order to discuss Cantors paper one should know it. Of > > course Cantor knew your trivial example and those proposed by Mr. > > Bader, who is seems to be used to take trivialities for the main issue. > > > > > Remove the line x=0 from > > > the plane. You remove an uncountable set and the result is clearly > > > disconnected. > > > > Yes, but Cantor considered only point sets which are dense. > > You did not state that above. I wrote about algebraic and transcendental numbers. So being dense is implied and need not be mentioned. It was only Mr. Bader who intermingled the sets of numbers. He is not even able to distinguish between rational and algebraic numbers, but as a typical crank, will never admit having committed an error. > > > "Betrachten > > wir irgendeine Punktmenge (M), welche innerhalb eines n-dimensionalen > > stetig zusammenhängenden Gebietes A überalldicht verbreitet ist und > > die Eigenschaft der Abzählbarkeit besitzt,..." > > But under *those* conditions there there are still uncountable sets that > leave the plane connected and also sets that leave the plane disconnected. Of course, but Cantor did not recognize it. And as far as I know nobody recognized it before I did. At least nobody mentioned it. Cantor tried to apply his result to the continuity of the physical space: He closed his paper by: "Daher liegt es nahe, den Versuch einer modifizierten, für Räume von der Beschaffenheit A gültigen Mechanik zu unternehmen, um aus den Konsequenzen einer derartigen Untersuchung und aus ihrem Vergleich mit Tatsachen möglicherweise wirkliche Stützpunkte für die Hypothese der durchgängigen Stetigkeit des der Erfahrung unterzulegenden Raumbegriffs zu gewinnen." He tried to explain the matter by countably many particles, the ether by uncountably many particles. Therefore the distinction between these both cardinals was very important for him. Regards, WM
From: mueckenh on 27 Nov 2006 09:32 Dik T. Winter schrieb: > In article <1164446451.353434.231240(a)45g2000cws.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > > > In article <1164126570.219569.222590(a)k70g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > ... > > > > For geometry, it is required, yes, for arithmetic it is not. > > > > > > And you do not think there is an underlying "something" to which set > > > theory with AC and without AC are related? > > > > No. > > So you think there is an underlyign "something" to which set theory with AC > and without AC are related. No, I do not think there is an underlyign "something" to which set theory with AC and without AC are related. > ... > > > > They have no representation which could be used for any form of > > > > Cantor's list. > > > > > > Well, according to set theory they have such representations. > > > > That is one of the reasons why set theory is wrong. > > Nothing more than opinion. You think there are no such representations, > so everything that states that there are such representations is wrong. I proved that there are no such representations. If they were (the paths in the infinite tree, for instance), then we had a contradiction. Regards, WM
From: mueckenh on 27 Nov 2006 09:43 Dik T. Winter schrieb: > > > No, there is *no* node in your tree that represents 1/3. Because there > > > is *no* node in your tree that represents an infinite sequence. On the > > > other hand, Cantor's diagonal proof is about infinite sequences. > > > > You do not require that one digit represents the number 1/3 in Cantor's > > list. You do not require that any digit there represents an infinite > > sequence. > > Of course not. > > > But you require that one digit (node) represents 1/3 in the tree? > > But you require that one digit (node) in my tree represents an > > infinite sequence? > > Of course. That is unfair. > > > Are you trolling, Dik? > > Not at all. > > > There is no node in the tree which represents 0.000... and no node > > which represents 0.1 But these real numbers are represented in the > > tree by paths. > > I do not understand. You state that the bits are the nodes. Yes. And an infinte sequence of bits is the representation of a real number. > I have > shown how you can extend that to the nodes representing numbers. I do no want not extend anything to nodes representing numbers. The paths represent numbers. > And > I also have shown how the numbers represented by the nodes all have > a finite sequence of bits. There is *no* node that represents 1/3. No. There is no digits in Cantor's list representing 1/3. > And when you switch to paths representing numbers you have something > completely different. I did not switch to paths but defined from the beginning that the paths represent the numbers. > Let's assign to a finite path the number of the > node where the last edge terminates. There is no last edge, because the paths are infinite, like the decimal representations used in Cantor's list. > Also in this case 1/3 is not in > your paths. Then 1/3 is not in Cantor's list. > There are no numbers assigned to the non-terminating > infinite paths. Then there are no numbers assigned to non-terminating decimal representations. > I think you wish that the edges represent the bits, > not the nodes. No, I do not wish that. (I could do so, but it would again confuse you. So we it as it is.) Regards, WM
From: mueckenh on 27 Nov 2006 09:49
Dik T. Winter schrieb: > In article <1164446537.991233.319900(a)j72g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > > > In article <1164126395.211430.7520(a)h54g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > ... > > > > 3) In order to show that there is no line containing all indexes of the > > > > diagonal, there must be found at least one index, which is in the > > > > diagonal but not in any line. This is impossible. > > > > > > I do not accept (3). It effectively states: > > > forall{n in N}thereis{m in N} (index m is not in line n) -> > > > thereis{m in N}forall{n in N} (index m is not in line n) > > > > For *finite, linearly ordered* sets, the quantifiers can be exchanged. > > Every line is a finite, linearly ordered set. > > But the sets are not the lines. The set considered is N, which is > linearly ordered but not finite. That N is the diagonal (and also the columns). But I do not consider them but the elements of *one* line (without fixing this line). The set of elements of one line (of each line) are linearely ordered and finitely many. Therefore I can state, for every line, that it is finite. Regards, WM |